I could use some advice (potential math or physics major)

So I was trying to sketch some plans about what to do in college. One option would be Engineering here in Brazil, in my city we have a great Engineering school, which is known for giving its students a ton of work opportunities - it wouldn't be too hard for me to get a fairly stable/wealthy life. But, after some soul-searching I realized I really like Mathematics and Physics and maybe I could succeed on one of that fields. Then I self studied some Calculus and Linear Algebra (no fancy AP courses here) and enjoyed it (MIT Open Course Ware was a huge help). Now, the thing is that, while I could study (almost 100% of getting accepted) one of these subjects here in what is known as the best university in Latin America (São Paulo University), I am seduced by the idea of studying in a top international university. Furthermore, after some searching, I found that my dilemma between Physics and Mathematics could be solved by doing some kind of hybrid course (double major), which seems possible in some universities. Is that reasonable?
However, we all know that getting admitted in MIT/Harvard/etc is pretty hard. While here in Brazil very few students learn Calculus, Linear Algebra and other college-level stuff in high school, this seems to be rather common in the US. And my application is probably not that much competitive - I haven't done any community service (and don't intend to - I don't have the time and I would be doing it merely for interest, which doesn't sound honest at all), I am not an amazingly creative and awesome person like everyone in MIT seems to be etc... And the big problem is that those colleges are very expensive (the São Paulo University here is a free public school)! If I do Physics or Math I won't be earning much (how poor are graduate students?), so I really cannot afford paying that much (even with financial aid the fees are still pretty darn expensive...).
Despite all these problems, I'll apply anyway. So which universities have the top Math or top Physics programs? Even better, which of them allows you to major in Physics, while having a strong mathematical background (allowing you to specialize in Mathematical/Theoretical Physics?).
I have done some research in the internet and MIT seems to fit what I search for - It seems possible to double major in Course 8/18 (Physics/Math). Then, I would take the Physics requirements alongside with mathematics like Abstract Algebra, a nice proof-based Analysis, an introductory Topology course and maybe Complex Analysis, applied math courses etc. I really don't know if that would be reasonable - so what are the mathematics needed for Mathematical/Theoretical Physics? Should I focus more on Pure or Applied math?
The Physics major in Caltech seems rather interesting since there are various undergraduate courses on math/theoretical physics and maybe I could fit in Abstract Algebra and Real Analysis (the double major doesn't seem possible).
I have heard well of Stanford, Harvard, Princeton etc.

Anyway, I am really sorry for having made you read this rant, but I could really use some advice on which universities have nice Maths/Physics programs around the world. If they happen to be less expensive, even better. And I would really like to know if you think that doing a double major / an mathematical physics program is a good idea or not (maybe it would be better if I focused in one of the fields since like it or not, they are related to each other so I would have to study physics in a math major and vice versa).

I'd suggest being as honest as can be in your application. In the event that you are not and you got in, then you got in for "wrong" reasons", which might interfere with your enjoyment at the said school. It could very well work out but MIT (not certain about the other schools) strives to build a community more than anything else. Another useful thing to do would be to try contact some Brazilians who got into MIT and other schools in North America. You should also note that these schools will consider applicants individually, meaning that - if I got this right - if you're from a bad high school or a country where certain facilities aren't available, they won't hold it against you. It could get problematic if you chose to *not do* things you could do.

A few persons here actually went to MIT and Harvard. They will have more sound advice to offer. Also, threads like yours are quite common in this forum, so searching a little might return some interesting results. You should also note that what I say is based on hours of reading and reading alone. I never went to any of these schools and will start my undergraduate degree next year or the year after that. I'm also not applying to college in the USA (anymore) for a few reasons.

On that note, you mentioned "around the world" and "less expensive". What languages can you speak? If you know any French or German, you could look into studying there. The French provide excellent education in their Classes Preparatoires Aux Grandes Ecoles, where if you choose the MPSI track, you will be doing ~9h of maths per week and ~5-7 (can't remember the exact number) hours of physics per week. It's a two year course, after which one should apply for and write exams in a "concours", where one will be competing against every student in the MP track who applied for that specific "concours". (there's a few different ones, granting access to quite a number of schools - some more flexible than others, in that they have more to offer in terms of what subjects one can study)

The other cool thing is you'll be doing lots and lots of maths and physics. It will also be more advanced than anything you'd find in a freshman class in the USA, expect perhaps the honours variant of intro classes offered at the likes of MIT or Harvard. The same could be said about Germany or the UK. Math majors, in general, start with Analysis. Calculus and intro differential equations are usually done in high school.

Also:
1) You don't have to commit to any specific subject in the first two years. You can choose to do either maths, physics, CS or engineering when you get to an ecole. :)
2) I've heard of a few Brazilians studying in the CPGE.

Edit:

Starting with more or less advanced courses could be appealing or not appealing, depending on the amount of work you've done in the previous topics and how comfortable you are with them.

Also, try not to get too sucked into the "it's Harvard/MIT/etc or it's nothing!" hole. It's a very dark place to be and you'd be lucky if someone *in the know* would appear out of nowhere and guide you out of it. By all means, apply if that's what you want.(also, look into Amherst and look for a thread I made about them, which contains some useful info from a forumer who went there; they too have an excellent financial aid package for international students) However, you should note that you don't absolutely need XYZ college in the USA (be it Minnesota or Yale) to get into a good graduate program. If you take a look at the CVs of grad students at Princeton's Math department, you'll find one or two from the University of Athens (!), among other places. Now, perhaps, Athens has a very active "math scene"....

So I did some research about those French grand écoles. I can speak some French, not very well, but if it is necessary, I can manage to improve it next year. From what I understood the écoles are graduate schools and the CPGE are something like a cram course to prepare you for them? It seems to be very stressful/competitive, with oral exams and so on... I get that a "CPGE lifestyle" is much more stressful than a undergraduate top American school lifestyle? And I assume the CPGE would be just as nice for enrollment in an academic career as would a grad school in [insert your top US school here] also be? A final thing I didn't understand... what are the costs for the CPGE/grand école?

If that is of any help, I can speak English, Portuguese (obviously), Spanish and a somewhat bad French.

So in those big European (Cambridge, Oxford, the French grand écoles) schools I am assumed to know everything about Calculus and Diff. Equations already from high school? Does every course start already at a very proof-based Analysis? If so, I could manage to systematically study those next year, but in the great American schools even the "honours" courses are not the same as the real upper division stuff (Analysis, Algebra, advanced Classical Physics etc.).

To be honest, the French system is a little complicated and the grandes ecoles could or could not be graduate schools. Following the Bologna process, what happened is, all participating countries started to uses a "common credits" system. So, for a Bachelor's degree, one needs 180 ECTS, for a Master's, anything between 90 and 120 and so forth. Generally, it's 60 credits per year, meaning the BSc (Licence in France) takes 3 years and the MSc takes 2 years. In England, however, they teach 90 credits worth of material in one year, in many cases.

You should note that the grande ecole system is *not* part of the university system. I'm not familiar enough with the history to give you a clear enough explanation though. If you're interested, fr.wiki would be a good place to start. As you must have found out, during your time in a CPGE, you will be in a separate section of a lycee. It's part of the lycee but separate in that all your teachers will be focusing solely on students in your specific track of the CPGE, i.e, they will not be teaching lycee students at all. The teachers, in many cases, have gone through the CPGE themselves and are usually educated to the M2 (5 years) level.

If you attend a public lycee, then education will be free. The grandes ecoles will vary from being free to being something in the order of 10k euros/year. When you're done with the CPGE, you can ask for equivalency (what your two years of work are worth in ECTS) and transfer to a university. What can be done is going into a "magistere" program, which is harder than the typical university program, in that it is designed for people who want to go into research. There's a few of them in Maths, Physics and CS as well, if I'm not mistaken. University students can also enter this track and it starts from the 3rd year of uni and ends at the 5th.
The more traditional route though, is to enroll into an ecole. The above was just to show you that there's other options.

You should also look out for the double degree programs. The ecoles have many agreements with other ecoles/universities, where by the time you graduate, you can have 2-3 degrees. Somebody actually posted on here, and was asking something about a postdoc at CalTech and he had Master's degrees in both Pure Maths and Physics. Actually, he might have had a third but I can't remember. Anyway, all of this was because of the double degree agreements of his ecole. He also had to put in some additional work.

In terms of preparation prior to entering a classe prepa, you should be able to comfortably do whatever they cover in the French Baccalaureate. It should probably be similar to a A-Level Mathematics (AS+A2) and Higher Level International Baccalaureate Mathematics. What they do in MPSI is a direct continuation of the former. http://prepas.org/mpsi [Broken] is their program. You might want to look into the PCSI track as well, if you like Chemistry. This looks like a sample paper but I'm not so certain. It actually looks a little harder than what I did in A-Level.

Yeah, being in a CPGE is quite stressful. I don't know how it is compared to X "top American college" and you'd only really know if you, yourself, try out both. You should however, expect to work yourself crazy if you want to get into a good ecole - competition is fierce. If you don't want to pursue a PhD by the time you're finishing the Master, being in an ecole with a good reputation/having good industry links would be a smart move. The good thing with the CPGE system is that if you don't get into the #1 ranked lycee for MPSI or PCSI, is that this is not bad news at all. You can enroll into any lycee which has an etoile class (MP*, PC*, etc) and you'd still be in practically the same position anyone at the #1 lycee will be. The only difference will be that those at the latter would have worked better during high school and competition within your class would probably be quite higher.

However, in the case of the UK (for example), the level of work being covered will decrease as you go down the league tables. I actually don't care much about rankings but as it turns out, the lower ranked universities offer much less advanced coursework than the higher ranked ones. Attending Imperial College to read mathematics is significantly different than attending Kingston to study the same subject, as is made evident by their respective syllabi. So in the case of the UK, doing maths/physics outside the top 15 (for e.g) could be very bad news. Matt Grime, an old poster here, at the time of the post was working at Bristol and he said that the depth covered there was significantly less than what he did at Cambridge. And Bristol is actually among the better English universities...

In a CPGE, everything across the board is standard. The only thing that changes is the concours that you're being prepared for. You can expect the MP* students being prepared for the more selective ecoles, like the Ecole Polytechnique or the ENS. It does however happen that somebody not in an etoile class is accepted into Polytechnique.

I know nothing about Portuguese and Spanish science education. Lavabug is studying Physics in Spain though, so he might have a thing or two to say.

a) admission:
Bluntly speaking, it is brutal for international students. That is absolutely not to say that you can't get in, just that you are facing extreme competition. However, the international students that I know have a very close-knit community and are definitely among the most brilliant students I have met.

b) 8/18:
This is a somewhat common combination, though not easy. I am not personally majoring in either but I am planning on taking classes in both. 18 is a short major to complete with relatively fewer requirements, and many classes in 8 and 18 overlap in terms of their material. I can't say much more without knowing your background; there is a wide range of students, from those who complete a single major in 8 or 18 and find it to be more than enough to those who skip out of many introductory requirements (often, internationals have the background to do this, by virtue of their prior achievements) and complete both 8 and 18 and go on to take advanced classes. There is a "flexible" variant of 8 that tends to allow for easier completion of a double major; the "focused" variant is generally viewed as more respected by graduate schools. It depends on what you'd like to pursue.

Definitely also consider Princeton if you are more math-leaning; Caltech if you are more physics-leaning. And, as Mepris mentioned, don't get too caught up in wanting to go to a "big-name" school. If you get in, all the more power to you, but if you don't, there is no shortage of opportunities in the world.

c) financial aid:
The international students I know come from a variety of backgrounds. I have heard some stories of MIT being stingy with financial aid for them, and to be fair, Ivy League schools (Princeton) are typically more generous as far as financial aid (you probably wouldn't have to worry there). However, there are ways of supporting yourself through work-study and other options and I wouldn't rule out a school based on financial obligations before applying.

The other thing to consider for MIT is that the amount of international students they do take in is around 150. That is not to say that they give out only 150 offers. A wise plan would be to save up some money (~$50-75 per school) and apply to a range of schools, concentrating one ones who would be more willing to offer need-based financial aid or scholarships (merit).
A cool way to save some money would be applying EA or ED. You get a decision much faster (by mid-December usually) and *if* you get in, you don't need to apply anywhere else. The tricky part with ED is that it's binding and I'm not certain as to how one can get out of this "agreement" in the event they're accepted and what the consequences are. At any rate, I'd be more comfortable to apply Early Action. Princeton, Yale and Harvard offer this to internationals as well, if I recall correctly.

Also, when you're considering the likes of Princeton or a CPGE (good grande ecole or magistere), you probably don't have much to worry with regards to the education you're receiving. At this point, what you should be concentrating your efforts on is how to make the most of it. The other thing to consider in the CPGE V/S American College "debate" is that in a CPGE, you are "doomed" to work yourself crazy for two years (it's very intense), whereas in college, if you burn out, you can always drop a class or two. Or not take the honours sequence. In a CPGE, you won't have a safety net. If you're not good enough in the first year, you're kicked out. Only the 2nd year can be repeated. You should also consider that in a CPGE, you will be faced with the stress of the concours throughout its duration and it's only when you finally enroll into a magistere, university program or a grande ecole that you could relax a little. If you attend college, you won't have to worry (too much) about "Damn! I can't perform in my orals. What am I gonna do?!" and it's only after 4 years (3, in case you graduate early) that you'll be going elsewhere. Although you'll probably apply to graduate school in your junior year.
Now, most people hate stress. No one wants to live in a constantly stressful environment. However, there's a minority who either naturally strive in this environment or like it for some reason. Figure out which one of the two guys you are. If you're among the minority, you will love a CPGE. However, you will also probably love it at a place like Harvard or MIT, where you have about 5 different intro mathematics classes to choose from. :-) :-) :-)

I say all of this because I wish I had gone out and read all about this when I was 15-16...